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Hovedindhold
Aktuel tid:0:00Samlet varighed:10:40

Tangent planes and local linearization

Video udskrift

- [Voiceover] Hello everyone. So I'm talking about how to find the tangent plane to a graph. And I think the first step of that is to just figure out how we control planes in three dimensions in the first place. So what I have pictured here is a red dot representing a point in three dimensions, and the coordinates of that point, easily enough are one two three. So the x coordinate is one, the y coordinate's two and the z coordinate is three. And then I have a plane that passes through it. And the goal of the video is going to be to find a function, a function that I'll call l, that takes in a two dimensional input x and y and this function l should have this plane as it's graph. Now the first thing to notice is that there's lots of different planes that could be passing through this point, right? At the moment it's one that's got a certain kind of angle, you could think of it going up in one direction, but you could give this a lot of different directions, and get a lot of different planes, that all pass through that one point. So we're going to need to find some way of distinguishing the specific one that we're looking at, which is this one right here, from other possible planes that can pass through it. And as we work through you'll see how this is done in terms of partial derivatives. But as we are getting our head around what the formula for this graph can be, let's just start observing properties that it has. The first property is that the graph actually passes through this point, one, two, three. And what that means in terms of functions over here, is that if you evaluated at the point one two, the input pair where both x is one and y is two, then it should equal three. It should equal three because that's telling you that when you go x equals one and y equals two and then you say what's the height of the graph above that point, it should be the z coordinate at the end of the desired point. So this right here is the fact number one, that we can take into consideration. And beyond that, let's start thinking about what makes planes, what makes these kind of flat graphs, different from the sort of curvy graphs that you might be used to in other multi-variable function. The main idea is that if you intersect it with another plane, so here I'm going to intersect it with y equals two, this kind of constant plane, and so I'll go ahead and write that down. That plane that you're looking at, is y equals two. And you can think of this as representing, what is movement in the x direction look like. As we move along the x direction, this kind of has a slope. The two planes intersect along a line, and that's one of the crucial features of a plane, is that if you intersect it with another plane, you just get a straight line, meaning the slope is constant, as you move in the x direction. But it's also constant, that same slope, if you move in the y direction. If I had chosen a different plane. If instead I had chosen y equals one, which looks like this, then you get a line with the same slope. And no matter what constant value of y you choose, it's always intersecting that plane, with a line that has the same slope. And now if you look back to the videos on partial derivatives, and in particular on how you interpret the partial derivative of a function, with respect to it's graph, what this is telling you is that when we take the partial derivative of l with respect to x, because constant y means you are moving in the x direction, this should just be some kind of constant. Some kind of constant a. I'll kind of emphasize that that is a constant value here. And the same goes in the other direction, right. Let's say instead of intersecting it with constant values of y, we say well what if you intersected it with a constant value of x, like x equals one. Well in that case what you should get, 'cos you're intersecting it with a plane, is another straight line, so these two planes are intersecting along a straight line, which means as you move in the y direction, this slope won't change. But also as you move in the x direction, if you imaginee slicing it with a bunch of different planes, all representing different constant values of x, you would be getting a line with that same slope. And that's telling you another powerful fact. The partial derivative of l, with respect to y, now if you're moving in the y direction, that's equal to some other constant, that I'm going to call b. And now, keep in mind, these are very powerful statements, because the partial derivative of l with respect to x is a function, this is a function of x and y, and that might actually be worth emphasizing here, that the partial derivative of x with respect to y is something that you evaluate at a point in two dimensional space. And we're saying that that's equal to some kind of constant value. Now that's a pretty powerful thing right? Because it's telling you, it's giving you control over the function at all possible input points, for movement in a specified direction. And the same goes over here. This is telling you that. A function is constantly equal to some value b, and we're not sure what this value b is. But just geometrically we can estimate what these things should be. So if I take back the plane representing a constant y value, and we say what's the slope, you're moving in the x direction, we've got a constant y value, what is the slope at which this plane intersects our graph? I would estimate this as about the slope of two. You know you go over one and it goes up two. So what that would tell you is that, at least in the specific graph we're looking at, this is at least approximately equal to two. And then similarly if we look at constant x value, and we say that represents movement in the y direction, what is the slope there. This looks to me like about one, as a slope. You kind of move over one unit you go up one unit. So I'd say down here that the constant value of the partial derivative with respect to y is about equal to one. So we have three different facts here, the value of the function at the point of one two, the value of its partial derivative with respect to x everywhere, and the partial derivative with respect to y, everywhere. And this information is actually going to be enough to tell us what the function as a whole should equal. Now specifically, this idea that the partial derivative with respect to x is constant, tells us that the function l of xy is going to equal two times x plus something that doesn't have any x's in it, something that as far as x is concerned is constant, because the only thing whose derivative, with respect to x, is the constant two is two x plus something that's constant, as far as x is concerned. And then similarly over here, if the partial with respect to y is the constant one, then that tells you that the whole function looks like this looks like a constant as far as y is concerned. So once we bring in y, it's going to be one times y plus something that's constant, as far as y is concerned. You know this part is already constant, as far as y is concerned, so anything that I add beyond here has to be constant as far as both x and y is concerned. So that part has to actually literally be a constant. So I'm just going to put in c for that, to represent constant. So, you can see this is a very restricted property on our function, because the only place x can show up is this linear term and the only place y can show up is as this linear term. And when I use the word linear, you can pretty much interpret as saying the term x shows up without an exponent or without anything fancy happening to it. It's just x times a constant, that's pretty much what I mean by linear. It's got more precise formulations in other contexts, but as far as we're concerned here, you can just think of it as meaning variable times a constant. So the question is what should this c be. And you can imagine that we can, based on this property, based on the value at the point one two, we can uniquely determine c. And you can plug in, x equals one, y equals two, know that this has to equal three and solve for c. Which we could do but I'm going to actually do something a little bit more convenient. I'm going to kind of shift around where the constants show up, and I'm going to say that the whole function should equal two times and then I'm going to put a constant in with the x, I'm going to say x minus one, and then I'm going to do the same thing with y, and I'm going to say plus one, here is the partial derivative with respect to y, y minus and then I'm going to say two. And the reason I'm doing this, notice this doesn't change the partial derivative information, it's just if we take the partial derivative with respect to x, this will still be two, and when we take it with respect to y, this will still be one. But the reason I'm putting these in here, is because we're gonna evaluate it at the point one two so I want to make it look as easy as possible to evaluate at the point one two. And then from here I'm just going to add another constant, so instead of saying c because this is going to be slightly different from c, I'll call it k. But the idea is that I'm just moving around constants. If you imagined distributing the multiplication here, and having two times that negative one, and one times that negative two, you're just changing what the value of the constant stuck on the end here is. Now the important part, the reason that I'm writing it this way, which is only slightly different, is because then when I evaluate this, at l of one two, this whole first part, cancels out because plugging in x equals one means this whole part goes to zero. Same with the second part, because when I plug in y equals two, this part goes to zero. So k this other constant, that I'm tagging on the end, is going to completely specify what happens when I evaluate this at the point one two. And of course I want it to be the case, when I evaluate at one two I get three. I want it to be the case when I evaluate at one two, I get three. So that tells me, that this constant k here should just equal 3. So notice the way that I've written the function here, is actually quite powerful. We have a lot of control. This term two was telling us the slope with respect to x. So when you moved purely in the x direction, and that was kind of illustrated here, purely in the x direction, that's telling us the slope with respect to x, and then this term one here, was telling us the slope with respect to y. So when we moved purely in the y direction, that's telling us the slope there. And we could just turn those knobs if we change the two and we change the one, that's what's going to allow us to basically change what the slopes of the plane are. I'm going to say slopes plural because it's with respect to the x and the y direction. And that will give us control of various different planes that pass through. If I'm looking at the one right here, then the movement in the y direction is very shallow, so that would be turning this knob lower, and instead of one it might be point zero one. And if I were looking at movement in the x direction, you know this looks actually negative. So this would tell you that it's going to be some kind of negative number. So you can kind of dial these knobs and that changes the different planes that pass through that same point. And then, plugging in this one two and three, tells us what point we're specifying. We're basically saying when you input x equals one, and you input y equals 2, the whole thing should equal three. So this form right here's is powerful enough that I want you to remember it for the next video. I want you to remember the idea of writing things down in this way where you specify the point its passing through with it's x coordinate y coordinate and z coordinate, placed where they are, and then you tweak the slopes, using these coefficients out front. So with that, I will see you next video.